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Sentences forApproximations of π
- In Proposition II, Archimedes gives an approximation of the value of pi, showing that it is greater than 223⁄71 and less than 22⁄7.
- converges linearly to, adding at least three decimals digits of precision per four terms, a pace slightly faster than the arcsine formula for :
- These representations led to fast algorithms for computing numerical approximations of.
- This means the scribe of the Moscow Papyrus used to approximate π.
- Tossing a needle 3408 times, he obtained the well-known approximation 355/113 for π, accurate to six significant digits.
- Standard measuring vessels dating to the reign of Wang Mang (9–23 CE) also showed approximations for pi at 3.1590, 3.1497, and 3.167.
- The list contains only formulae whose significance is established either in the article on the formula itself, the article Pi, or the article Approximations of.
- Pi Approximation Day is observed on July 22 (22/7 in the day/month format), since the fraction 22⁄7 is a common approximation of, which is accurate to two decimal places and dates from Archimedes.
- For more detailed explanations for some of these calculations, see Approximations of.
- In the 14th century, Madhava of Sangamagrama, the founder of the Kerala School of Mathematics, found the Madhava–Leibniz series and obtained from it a transformed series, whose first 21 terms he used to compute the value of π as 3.14159265359.
- He was an expert in the field of dynamics and the approximation of.
- By analogous methods to non-standard analysis one can also use PA to define close approximations to irrational multiples of a non-standard number u such as the least v with v > ⋅ u (these can be defined in PA using non-standard finite rational approximations of even though pi itself cannot be).
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